Secondary Interaction Contribution in Hydrogen-Bonded Complex:
Theoretical Model Study in Hydrogen Fluoride Trimer

Shun-ichi KAWAHARA and Tadafumi UCHIMARU


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1 Introduction

Molecular recognition using a plurally hydrogen-bonded complex, for example, base pair formation in a nucleic acid duplex (Figure 1) [1], is an important topic in supermolecular chemistry [2]. It has been widely studied theoretically [3]. In such complexes, the arrangement of the proton donor (D) -acceptor (A) orientation has an important role in the total stability of the complex. Jorgensen et al. proposed secondary interactions in plurally hydrogen-bonded complexes [4,5]. The secondary interactions are defined as follows. The D-A type secondary interaction: an interaction between an acidic proton, forming a hydrogen bond, and a neighboring proton acceptor heavy atom. For example, an interaction between the 1-position proton of guanine and the exo-cyclic 2-position oxygen of cytosine. The A-A type secondary interaction: an interaction between a proton acceptor heavy atom and a neighboring proton acceptor heavy atom. For example, an interaction between the 1-position nitrogen of adenine and the exo-cyclic 4-position oxygen of uracil. The D-D type secondary interaction: an interaction between an acidic proton, forming a hydrogen bond, and a neighboring acidic proton. For example, an interaction between the 3-position proton of uracil and the exo-cyclic 6-position amino proton of adenine. Whether or not the neighboring hydrogen bond site forms a hydrogen bond, the secondary interaction should be considered; the A-A type secondary interaction should be considered to be between the exo-cyclic 2-position oxygen of uracil and the 1-position nitrogen of adenine, for example.


Figure 1. Watson-Crick type base pair and hydrogen bond arrangement in a plurally hydrogen-bonded complex.

According to Jorgensen's proposal, the total hydrogen bond should be more stable when all the proton donor-acceptor pairs are arranged in the same direction as in Figure 1 I: DD-AA type complex, for example, the arrangement of the hydrogen bonds B and C in a guanine - cytosine base pair. On the other hand, the total hydrogen bond should be less stable when the proton donor-acceptor pairs are arranged alternately as in Figure 1 II: DA-AD type complex, for example, the arrangement of the hydrogen bonds A and B in guanine - cytosine and adenine - uracil base pairs. As a result, the adenine - uracil base pair is considered to have two hydrogen bonds and two repulsive secondary interactions (A-A and D-D type), and the guanine - cytosine base pair is considered to have three hydrogen bonds, two attractive secondary interactions (D-A type), and two repulsive secondary interactions (A-A and D-D type) [6]. We already reported the importance of secondary electrostatic interactions in hydrogen-bonding complexes [7], and we report herein the contribution of the secondary interactions to the whole hydrogen bond formation energy, using hydrogen fluoride dimer/trimer as the models of the hydrogen bond site of a plurally hydrogen-bonded complex.

2 Computational Methods

The molecular interaction energies were evaluated by a supermolecular method. The basis set superposition error (BSSE) for hydrogen bond energies was corrected by using the counterpoise method [8]. In the supermolecular method, only total interaction energy can be calculated, and it is difficult to distinguish between the hydrogen bond energy and secondary interaction. A molecule, which has plural hydrogen bond sites (see formamide in Figure 2, for example), should be divided into each hydrogen bond site. Hydrogen fluoride was employed as a model of the divided hydrogen bond sites.
For the estimation of the molecular interaction, B3LYP/6-311+G**, MP2/aug-cc-pVTZ, and QCISD(T)/aug-cc- pVTZ levels of the calculations were applied. The structure of the hydrogen fluoride was optimized in B3LYP/6-311+G** for the DFT calculations (RHF = 0.9222 A) and in QCISD/aug-cc-pVTZ (RHF = 0.9190 A) for the MP2 and QCISD(T) calculations.
The hydrogen bond lengths in B3LYP (2.800 A), MP2 (2.800 A), and QCISD(T) (2.900 A) studies were determined by the potential energies calculated at the levels of B3LYP/6-311+G** and QCISD/aug-cc-pVTZ using the linear (C+v) structure [9]. Only in the MP2 study, the basis set limit was estimated with Feller's procedure [10] using aug-cc-pVDZ, aug-cc-pVTZ and aug-cc-pVQZ basis sets. The distance between the hydrogen fluorides was fixed to 2.25 A (Figure 2), considering the distance between the hydrogen bond sites of the typical molecules forming a plurally hydrogen-bonded complex, e. g., formamide dimer and nucleic acid base pairs.
We evaluated the secondary interactions using two types of models: hydrogen fluoride dimer models, and hydrogen fluoride trimer models. In the hydrogen fluoride dimer systems, hydrogen bond energy and secondary interaction energies were evaluated by the arrangement of the hydrogen fluorides: Figure 2 I; for the hydrogen bond energy, Figure 2 II; for the D-A type attractive secondary interaction, Figure 2 III; for the D-D type repulsive secondary interaction and Figure 2 IV; for the A-A type repulsive secondary interaction.


Figure 2. Modeling for divided hydrogen bond sites of plurally hydrogen-bonded complex by the arrangement of the hydrogen fluoride dimers/trimers.


Figure 3. Estimation of the hydrogen bond (x), the repulsion of neighboring HF (y) and the secondary interaction (z) for the model of neighboring hydrogen bond sites by the combination of hydrogen fluoride trimer division (l, m and n).

Figure 2 V-VII show the arrangement of three hydrogen fluorides for the D-A type attractive secondary interaction (Figure 2 V), the D-D type repulsive secondary interaction (Figure 2 VI) and the A-A type repulsive secondary interaction (Figure 2 VII). The magnitude of each interaction was evaluated as follows. The hydrogen fluorides were named as shown in Figure 3. D-A type arrangement of three hydrogen fluorides (Figure 2 V) is shown for example. Three molecular interaction energies should be considered in the system: x = hydrogen bond interaction between HFA and HFB (EHB), y = repulsive interaction between HFA and HFC (Erep), and z = secondary interaction between HFB and HFC (E2). However, these interactions could not be calculated directly by the supermolecular method; thus, to determine these three interactions, three interaction energies, i. e., l: total stabilization energy of (HFA-HFC)+HFB (interaction energy between (HFA-HFC) and HFB, here, HFA-HFC were dealt as one part. See also Figure 3), m: interaction energy of (HFB-HFC)+ HFA and n: interaction energy of (HFA-HFB)+HFC were calculated. The interaction l is considered to include the interactions x and z, the interaction m is considered to include the interactions x and y, and the interaction n is considered to include the interactions y and z; thus, l = x + z, m = x + y, and n = y + z. D-D and A-A type secondary interactions were also calculated as above, using the corresponding trimer complex (Figure 2 VI and VII, respectively)
All molecular orbital calculations were carried out using a Gaussian 94 program [11].

3 Results and Discussion

Tables 1, 2 show the hydrogen bond energies and three types of secondary interaction energies, using the hydrogen fluoride dimer systems (Figure 2 I-IV) and the hydrogen fluoride trimer systems (Figure 2 V-VII), respectively, in the B3LYP/6-311+G** level of the calculation. Similar results, the ratio of the secondary interaction based on the hydrogen bond, were observed in both dimer systems and trimer systems. The D-A type attractive secondary interaction was evaluated as 20-30% of the hydrogen bond (I in the dimer sytem and x in the trimer system). Unexpectedly, the D-D type secondary interaction was evaluated as weakly (4-5%) attractive in this level of theory. The A-A type repulsive secondary interaction is about 20% of the hydrogen bond. Similar results were observed based on the hydrogen fluoride dimer systems and the hydrogen fluoride trimer systems in the DFT calculations.

Table 1. Molecular interaction (the hydrogen bond and the secondary interaction) energies (kcal/mol) and the ratio of the secondary interaction based on the hydrogen bond stability by use of hydrogen fluoride dimer models, in B3LYP/6-311+G**.
EnergyRatioModel in Figure 2
EHB-3.82-I
E2 D-A-1.070.28II
E2 D-D-0.140.04III
E2 A-A+0.770.20IV

In the trimer systems, three-body interaction should be considered [12]; however, three-body interactions in the D-A type, the D-D type and the A-A type trimer (ET = Etotal - (EHB + E2 + Erep)) were all small, comparing total interaction energies of the complexes (Etotal = EHFtrimer - 3EHFmonomer). Rincon et al. reported hydrogen bond cooperativity in the hydrogen fluoride clusters. The stabilization energy per one hydrogen bond in the cluster significantly arises from the tetramer, but it is not remarkably changed between the dimer and the trimer [13]. Our results were in good agreement with the results of Rincon et al. [13]; thus, we can safely deal with these trimer systems as the model of hydrogen bond sites without consideration of three-body interaction. Taking account of these results, MP2and QCISD(T) studies were carried out based on the hydrogen fluoride dimer systems. Table 3 shows the hydrogen bond energy and three types of secondary interaction energies in various methods using the hydrogen fluoride dimer systems. The basis set effect in the MP2 method was small. The D-A type secondary interaction was estimated to be about 30%. The result of D-A type attractive secondary interaction was similar to that of DFT. The D-D type secondary interaction was evaluated to be about 10% (QCISD(T)) or about 15% (MP2) attractive. The A-A type repulsive secondary interaction was about 10-15% of the hydrogen bond. Thus, DFT estimated D-A type secondary interaction in high accuracy, but it underestimated the contribution of the D-D type and overestimated the A-A type secondary interaction.

Table 2. Molecular interaction (the hydrogen bond and the secondary interaction) energies (kcal/mol) and the ratio of the secondary interaction based on the hydrogen bond stability by use of hydrogen fluoride trimer models, in B3LYP/6-311+G**.
Interaction EnergyInteraction in Figure 3Ratio(z/x)
EHB-3.52-3.85-3.85x-
Erep++11.64+5.13+5.13y
E2 D-A-0.79z0.22
E2 D-D-0.19z0.05
E2 A-A+0.72z0.19
ET-0.29+0.05+0.04three-body interactions
Etotal+7.04+1.13+2.04total interaction
VVIVIIModel in Figure 2
+Repulsive interaction between HFA and HFC in Figure 3. See the computational method in the text.


Figure 4. Potential energy curves of the A-A type (cross), DD type (circle), D-A type (plus) secondary interactions, and hydrogen bond (square) depending on the hydrogen bond length (between the heavy atoms). See also Figure 2.

Figure 4 shows the change in the secondary interaction depending on the distance between the hydrogen fluorides, which corresponds to the hydrogen bond length, in DFT (A) and QCISD(T) (B) methods. The trend of these two figures is similar: all interactions in DFT are a parallel shift of the results in QCISD(T). The D-A type and the A-A type interactions were quite insensitive to the hydrogen bond length. In contrast to these two interactions, the D-D type interactions were sensitive to the hydrogen bond length. It was repulsive when the distance between the hydrogen fluorides was long, and it was attractive when the distance between the hydrogen fluorides was short. As the result, it was weakly attractive in the range of normal hydrogen bond length. From the result of this potential energy study, Figure 5 is considered to be the reason why the D-D type secondary interaction was evaluated as attractive: there are two tertiary attractive interactions between the proton and the fluorine atom of the opposite side, and these two interactions overcome the repulsive interaction between two protons (Figure 5A) [14]. In the case of the D-D interaction, tertiary interaction is 52 degrees bent hydrogen bond-like interaction. The atom distance between the hydrogen and the fluorine (RHF = ca. 2.2A) was comparable with the atom distance in the hydrogen bond (1.8 A). On the other hand, the RHF in A-A type arrangement (Figure 5B) was much longer (ca. 4.0 A). The fluorine-hydrogen-fluorine bent angle (ca. 149 degrees ) is too highly bent to form the hydrogen bond-like interaction. Thus, tertiary interaction in A-A type arrangement is considered to be negligible.

Table 3. Molecular interaction (the hydrogen bond and the secondary interaction) energies (kcal/mol) and the ratio of the secondary interaction based on the hydrogen bond stability by use of hydrogen fluoride dimer models, in various calculation levels.
B3LYP/6-311+G**MP2/aug-cc-pVTZMP2/basis set limit+QCISD(T)/aug-cc-pVTZ
EnergyRatioEnergyRatioEnergyRatioEnergyRatio
H-Bond-3.82--3.24--3.59--3.35-
2nd D-A-1.07-0.28-1.08-0.33-1.08-0.31-1.07-0.32
2nd D-D-0.14-0.04-0.48-0.15-0.51-0.14-0.30-0.09
2nd A-A+0.77+0.20+0.47+0.15+0.43+0.12+0.44+0.13
+: The values calculated based on the basis set limit, estimated by Feller's procedure [10].

From the results in Figure 4, both D-A type and A-A type secondary interaction energies were almost independent of the hydrogen bond length. In contrast to these two types of secondary interaction energies, D-D type secondary interaction energies were highly dependent on the hydrogen bond length. It was repulsive in the case of longer hydrogen bond length, and it was attractive in the case of shorter hydrogen bond length. Thus, the hydrogen bond length of the DA-AD type complex should be shorter than that of the DD-AA type, if the hydrogen bond capability of each hydrogen bond sites is the same. On the other hand, total hydrogen bond stabilization in the DD-AA type complex should be larger than that of the DA-AD type complex.


Figure 5. The tertiary interaction of D-D/A-A arrangement.

Next, a model study using a formamide-hydrogen fluoride complex was attempted. There are two proton donor (D1 and D2 in Figure 6) sites and acceptor (A1 and A2) sites in formamide. The hydrogen bond sites A1 and A2 are proton acceptor sites without and with D-D type secondary interaction. The hydrogen bond sites D1 and D2 are proton donor sites without and with A-A type secondary interaction. The proton donatability of D1 and D2 sites, and acceptability of A1 and A2 sites without the secondary interaction should be similar, respectively. Thus, the contribution of the secondary interaction should be observed as the difference in the hydrogen bond stability.


Figure 6. Hydrogen bond sites of formamide.

Table 4 shows the results in B3LYP/6-311+G** and MP2/aug-cc-pVTZ levels of the calculation. In both calculation levels, change in the stabilization energy with and without the secondary interaction was in good agreement with the results from the model complex using hydrogen fluorides (compare the ratio of the secondary interactions based on the hydrogen bond energy of Tables 3, 4). D-D type secondary interaction was a very small attractive force; on the other hand, A-A type secondary interaction was a large repulsive force, and the ratio of the A-A type secondary interaction in the total interaction was 0.2-0.3. Thus, the secondary interaction contribution in a plurally hydrogen-bonded complex, shown in Table 3, is applicable for rough estimation of the trend in stabilization of the plurally hydrogen-bonded complex.

Table 4. Total interaction and the secondary interaction energy (kcal/mol) and the ratio of the secondary interaction in formamide-HF complex.
B3LYP/6-311+G**MP2/aug-cc-pVTZ
H-Bond SiteTotal Int.2nd Int. (ratio/type)Total Int.2nd Int. (ratio/type)
D1-2.37-2.30
D2-1.700.67 ( 0.28 /A-A)-1.760.54 ( 0.24 /A-A)
A1-9.76-9.68
A2-9.730.03 ( 0.003 /D-D)-9.330.35 ( 0.04 /D-D)

4 Conclusion

The contribution of the secondary interaction in a plurally hydrogen-bonded complex was estimated using model complexes. Donor-acceptor type secondary interaction was attractive, and it had the largest (about 25-35% based on hydrogen bond interaction) contribution. Unexpectedly, donor-donor type secondary interaction was attractive, but the interaction was relatively small or, in some cases, negligible. Acceptor-acceptor type secondary interaction was strongly repulsive (about 10-20% based on hydrogen bond interaction). The estimation of the secondary interaction contribution was in good agreement with the change in hydrogen bond interaction energies in each formamide hydrogen bond site. Thus, the secondary interaction contribution in a plurally hydrogen-bonded complex, described here, is applicable for rough estimation of the trend in stabilization of the plurally hydrogen-bonded complex.

The services and computational time made available by the National Institute of Advanced Industrial Science and Technology (AIST) have been essential to this study and are gratefully acknowledged.

References

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[14] The secondary interaction in this study is the interaction between two hydrogen bond formation atoms, located at neighboring hydrogen bond sites (H-H for D-D type andF-F for A-A type in Figure 5). We referred to tertiary interaction as the interaction between an atom, which forms a hydrogen bond directly (H and F in Figure 5), and an atom, which does not form a hydrogen bond directly (hydrogen and fluorine atoms given in blanked boldface characters in Figure 5).


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